Momentum and Impulse

How hard something is to stop, $p = mv$

Momentum captures how much "oomph" a moving object has, combining how heavy it is with how fast it moves, and that is what makes it hard to stop.

Momentum is the product of an object's mass and velocity, given on the NESA formula sheet by:

It is a vector, pointing in the same direction as the velocity, measured in $\text{kg m s}^{-1}$. A heavy slow truck and a light fast bullet can have similar momentum. Because momentum is a vector, direction matters, a $5\ \text{kg m s}^{-1}$ momentum to the right is different from one to the left, which becomes vital in collisions.

Momentum is $p = mv$, a vector in the direction of the velocity, measured in $\text{kg m s}^{-1}$. It combines mass and speed into a single measure of how hard an object is to stop; direction matters because momentum is a vector.

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A force acting over time changes momentum, and longer times mean gentler forces

We just defined momentum as $p = mv$. That raises a question: what does it take to change an object's momentum? This card answers it → an impulse, a force applied over a time, equal to the change in momentum: $Ft = \Delta p$.

To change momentum you must apply a force for a length of time, and the same momentum change can come from a big force briefly or a small force for longer.

Impulse is the change in momentum, produced by a force acting over a time interval. On the NESA formula sheet:

Rearranging gives $F = \dfrac{\Delta p}{t}$: for a fixed change in momentum, a longer impact time $t$ means a smaller force $F$. This is exactly why airbags, crumple zones, and bending your knees when landing all reduce injury, they extend the stopping time.

Impulse equals the change in momentum: $Ft = \Delta p = mv - mu$. Since $F = \Delta p / t$, a longer impact time gives a smaller force. Catching the ball in $0.020\ \text{s}$ needs $225\ \text{N}$, but giving with it over $0.10\ \text{s}$ needs only $45\ \text{N}$.

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Newton's third law guarantees the total momentum is unchanged

We just used impulse to change one object's momentum. That raises a question: when two objects collide, why does their total momentum stay the same? This card answers it → because they push on each other with equal and opposite forces for the same time (Newton's third law).

In a collision, the two objects push on each other equally and oppositely, so whatever momentum one gains, the other loses, the total is unchanged.

The condition for momentum conservation is that the system is isolated, no net external force acts on it. During a collision, the two objects exert equal and opposite forces on each other (Newton's third law) for exactly the same contact time. So the impulse on object A is equal and opposite to the impulse on object B: $F t$ on one equals $-F t$ on the other. Therefore the momentum gained by one is exactly the momentum lost by the other, and the total momentum is unchanged:

This holds for any collision, whether the objects bounce apart or stick together, as long as no outside force interferes.

Momentum is conserved when no net external force acts (an isolated system). In a collision the objects exert equal and opposite forces for the same time (Newton's third law), so the impulses are equal and opposite, one gains exactly the momentum the other loses. Thus $\sum p_{before} = \sum p_{after}$.

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